First, I would like to apologize if this is a silly question. I know that
$$H^1 = \{f \in L^2 : f' \in L^2\}$$
My question is: for to check that a certain function belongs in $H^1$ it's suffices to show that $f'$ belongs in $L^2$? That is, exists a function outside $L^2$ such that whose (weak) derivative is in $L^2$?
In unbounded domain, this is false. In dimension $d$, if the function has $f'\in L^2$ and is converging to $0$ at infinity, then by Sobolev embedding it is in $L^p$ for some $p> 2$.
In particular, in bounded domains, it is true since them $L^p\subset L^2$.